I'm sure you can get very radical, with the whole silence thing, but from a "main-stream build" perspective ... It is possible to build a VERY powerful PC that has less than 20dB noise rating, on any moving part ...
SilverStone makes a few PSUs that Rate under 20 dB ... There are many others.
A 120mm PSU fan rated under 20dB is a good place to start ... get the highest quality, tho.
CASE FANS: Several are available which do greater than 30CFM (flow) @ under 20dB
SSDs ... You don't SAY !?
GPU is a weak spot ... Passive cards are mostly for HTPC ... not SLI/gaming ...
... There are a few very elaborate and expensive water cooling setups but, let's not get too freaky. I would look (CLOSELY) at two 9800GT-EE (low power) cards (not at newegg).
HDD ... Seagate 7200.12 series drives ... if a spinner is in the build.
CASE ... Bigger is actually better ... not the kind with "screen-porch" venting (unless 100% passive!) ...
... Placement ... How about running all the cables, thru the wall, from the next room ? ... I have lined the "computers-cubby", under my studio console, with 2-foot square acoustic ceiling tiles with carpet remnants (door mats, actually) glued onto them ... Like a 5-sided acoustic cube. The rear one is movable for cable routing and venting ("lean-to" style)
... Room Treatment ... Suffice it to say ... The room acoustics are important and they are under YOUR control. I have a long term project to get carpet remnants and surplus acoustic tiles from our local Habitat (For Humanity) "Re-Store" wher contractors get credit for surplus donations of unused/re-used materials.
Or ... just get a good set (or two) of fully-isolated, noise-cancelling studio monitor headphones ... and be done with it.
= enough! =